Significant figures summing

Table 6-1 lists the experimental quantities, k, T, ct, the transformed variables x, y, and the weights w. (It is necessary, in least-squares calculations, to carry many more digits than are justified by the significant figures in the data at the conclusion, rounding may be carried out as appropriate.) The sums required for the solution of the normal equations are... 

Note As expected, the percentages sum to 100%, within the limits of significant figures. 

Total score = sum of parts a-d. If any of the final answers has the incorrect number of significant figures, subtract one point. 

Estimates are based on a wet soil density of 1.8 tons/yd (2.1 tons/m ) based on SITE demonstration results. Costs based on contaminated soil treated. AU costs are rounded to two significant figures, based on the sum of the individual costs before rounding. 

Express the following sum with the proper number of significant figures ... 

The rule for addition and subtraction concerns the position of the last significant figure in the sum—that is, the location of this figure relative to the decimal point. The rule is When two or more numbers are added or subtracted, the positions of the last significant figures of each number relative to the decimal point should be compared. Of these positions, the one farthest to the left is the position of the last permissible significant figure of the sum or difference. 

Also notice that Figure 3 shows that the sum of the mass of the chlorine molecule and the mass of the phosphorus trichloride molecule is slightly smaller than the mass of the phosphorus pentachloride molecule. This difference is the result of rounding off and of correctly using significant figures. 

All masses and molar masses contain four significant figures. Therefore, the percents are correctly stated to four signigicant figures. The sum of the mass percents is 100.00% as required. 

All values have a minimum of three significant figures, so the mass of P4O10 is correctly stated with three digits. The mass of excess O2 (17.7 g) is found by subtracting two numbers that are accurate to the first decimal place. Therefore, the mass of excess O2 correctly shows one decimal place. The sum of the oxygen that was consumed (32.3 g) and the given mass of phosphorus (25.0 g) is 57.3 g, the calculated mass of the product phosphorus decoxide. 

These features of the VB approach, however, are preserved only under certain conditions . One of them is the direct treatment of the non-orthogonality, which becomes important when the separation between the systems decreases to the sum of the van der Waals radii and below. The approach is based upon the variation theorem, and consequently the interaction energy is computed as the difference between two large numbers. This difference is about 10 , and therefore very high accuracy is required in the computation in order not to lose all significant figures. However, under certain conditions this cancellation can be entirely avoided (see below). 

In general, it may not be possible to specify the value of the sum, S, in terms of Instead, we chose a value of and compute the sum to a given number of significant figures or decimal places. 

In addition and subtraction, the number of significant figures to the right of the decimal point in the final sum or difference is determined by the smallest number of significant figures to the right of the decimal point in any of the original numbers. Consider these examples ... 

Helpful Hint Note that the sum of the percentages to four significant figures should be 100% ... 

As a check of the results, note that the sum of the individual results, 82.8% and 17.2%, equals 100.0%, to four significant figures. 

The column labeled K sum is the term in large parentheses, which includes the inner summation. The J sum converges (to 4 significant figures) only at about J = 80 the K sum converges much more quickly. But the sum fails to take into account nuclear statistics, so it must be divided by the symmetry number (cr = 3). At 298 K, qR = 2.50 x 103. A similar computation at T = 500 K yields =... 

The 120.1 limits the sum to one decimal place. Remember, in addition the number with the fewest decimal places determines the number of significant figures in the answer. 

One must be aware that the accuracy of a computation is limited by the number of significant figures with which numbers are stored in the computer. Since one cannot completely avoid small differences of large numbers, one looses a few figures anyway. It is convenient to impose a threshold defining the final accuracy that one wants to have. Quantities, say in a sum, which... 

Theoretically, the final volume of liquid equals 380.6 ml but since the uncertainty in the value of the graduated flask is so high, it is unreasonable to carry four or even three significant figures. Thus, the best estimate of the combined volume is (380 13) ml. (Note that the uncertainty in the total volume is the square root of the sum of the squares—the variance—of each measure.)... 

The sum converges rapidly to 4.121 x 10 as can easily be verified numerically in fact, to three significant figures, terms after the first do not affect the sum. So the second-order correction is... 

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